Behaviour of L-Estimators of Location from the Point of View of Large Deviations

Abstract

Let X₁, … , X_n be a random sample from a population with density f(x-Θ) such that f(x) is symetric and positive. It is proved that the tails of the logarithmic derivative of the density of L-estimators of Θ converge at most n-times faster than the tails of the logarithmic derivative of the basic density and, on the other hand, there are estimators which behave from this point of view in the same way as one single observation. It is shown that both extreme cases may happen for the sample mean. Moreover, behaviour of some typical L-estimators of Θ is studied from this point of view.